Re: Continuous Functions

Is that really undefined at x = 1 ? I'm not sure about the answer to that.

You can say it isn't defined there in the definition, or you could define it to be 2, but it's unclear.

There are lots of situations where we simplify and then substitute after (eg. calculus). This function obeys the continuous definition requirement at x = 1. I accept the point about 0/0 but this example may nevertheless cause controversy amongst your readers.

Suggestion:

Replace with

Bob

Children are not defined by school ...........The FonzYou cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Re: Continuous Functions

Hi bob bundy;

I have asked that question too, why can't I simplify? I was told you can not simplify there because when you do you have changed the domain of the function. So the original function and the simplified one are not the same.

In mathematics, you don't understand things. You just get used to them.If it ain't broke, fix it until it is.No great discovery was ever made without a bold guess.

Sure, it's very contrived but it demonstrates that it's all in the definition.

eg y = (x^2-1)/(x-1) for all x in {reals} except x = 1

y = 2 when x = 1

This function is continuous for all x.

I'm happy that it stays unchanged; but it may provoke this discussion again. Maybe that's a good thing ??

bobbym wrote:

changed the domain of the function

There have been several posts recently that seem to suggest that once you have been told the equation for the function then the domain follows of its own accord. I disagree. I think that the domain is part of the definition and hence it's up to the definer to declare whether the domain is changed.

eg.

y = (x^2-1)/(x-1) for all x in {reals} except x=1

y = x+1 when x = 1

Bob

Children are not defined by school ...........The FonzYou cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Re: Continuous Functions

Is that really undefined at x = 1 ? I'm not sure about the answer to that.

You can say it isn't defined there in the definition, or you could define it to be 2, but it's unclear.

There are lots of situations where we simplify and then substitute after (eg. calculus). This function obeys the continuous definition requirement at x = 1. I accept the point about 0/0 but this example may nevertheless cause controversy amongst your readers.

Suggestion:

Replace with

Bob

Those functions are not continuous, but the sinvularities they have are removable. Try searching removable singularities. I think there is a Wikipeadia article on it.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Continuous Functions

Re: Continuous Functions

Hi Bob

You're welcome.

Hi MIF

Awesome page!

Last edited by anonimnystefy (2012-11-08 03:01:12)

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Continuous Functions

Hmm, I'm honestly a little bit confused about 1...indirect thing. Is 0/0 really undefined? I would have honestly thought it equals 0. I'm a little bit confused on that one part, so anyone care to explain (why 0/0 is undefined)? Though I do understand why it's not a continuous function. Other then that, I'd say you explained things clearly enough to be able to understand everything, so I'd say you did a pretty good job.

As for one of the posts, I don't really understand the 2=1 proof. Everything seems right until you get to a+b=b, nor do I understand how they got to that. I fail to see how this is proving 2=1, or rather, making it arguable I should say.

There are always other variables. -[unknown]But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -AristotleEverything makes sense, one only needs to figure out how. -[unknown]

Re: Continuous Functions

If you are talking about the 3'rd line where it says, (a-b)(a+b) = b(a-b), you are multiplying, not dividing, unless I'm missing something...?

Also, I'm not sure I quite understand the reason for simplifying it. Everything seems fine about it too me...

There are always other variables. -[unknown]But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -AristotleEverything makes sense, one only needs to figure out how. -[unknown]

Re: Continuous Functions

If you are talking about the 3'rd line where it says, (a-b)(a+b) = b(a-b), you are multiplying, not dividing, unless I'm missing something...?

Let a=ba² = aba²-b² = ab-b²(a-b)(a+b) = b(a-b)a+b=b1+1=12=1

In line three you have a factor of (a-b) on both sides. On line 4 you do not. Both sides were divided by (a-b) which is equal to 0. That is the misstep. Generally, you never divide by a variable unless you are sure that it cannot be 0. Sometimes such divisions are fatal, sometimes they just destroy answers. For instance

Divide both sides by x

Here the decision to divide by x, has cost us the other root which is 0.

In mathematics, you don't understand things. You just get used to them.If it ain't broke, fix it until it is.No great discovery was ever made without a bold guess.

Re: Continuous Functions

Hi!

Here's another explanation why division by zero is not possible. Division is a secondary operation,not a primary operation; that is, division is not given in the field axioms that establish the realnumber system. Multiplication is given in the axioms (as well as addition, but not subtraction).Division is then defined as multiplication by reciprocals.

For example suppose x is a real number whose reciprocal is y. Then given a number z we definez/x as z*y. So to divide by x, x must have a reciprocal. BUT zero has no reciprocal as stated in the multiplicative inverse axiom: For each real non-zero number x, there is a real number y suchthat xy=1. Of course the usual notation for the reciprocal of x is written 1/x. But this looks likedivision, so using 1/x for the reciprocal in the axiom and then later using "/" for division is a bitconfusing and makes the definition via z/x = z*(1/x) look circular.

When all is said and done, the axioms do not allow for a reciprocal of zero, hence division by zerois a non-issue --- it could never happen since zero has no reciprocal to define division by zero interms of.

Having zero in the denominator of a fraction is a similar issue. The set of fractions based on theset of whole numbers W={0,1,2,3,...} is defined as

F = { p/q | p and q are in W and q is not zero }

If someone asks why we can't have something like 2/0 the answer is simply the definition doesNOT allow zero in the denominator. It has nothing to do with "division by zero."

Analogously the field axiom for the reals do not allow zero to have a reciprocal. Hence to writean expression one would read as "division by zero" simply violates the field axioms. Trying towrite "a number divided by zero" would be defined as "a number times the reciprocal of zero."But a "reciprocal of zero" does not exist.

The other explanations of why one can't divide by zero illustrate the problems that would occurif we did allow zero to have a reciprocal. As such they provide good reasons from disallowingzero to have a reciprocal in the first place.

A nice physical example of trying to divide by zero can be seen in the operation of the old mechanical calculators of yesteryear. They operated on division as a "repeated subtraction."If one tried to divide 2 by zero, the calculator would subtract zero from two, add one to thequotient, and then check to see if there was enough left to subtract zero from it again. Of coursethere was enough left to subtract zero again since 2>0. So it subtracted zero from two again,added one to the quotient, and checked to see if there was enough left to subtract zero again.The old mechanical calculators got stuck in an "infinite loop" subtracting zero over and overagain. The quotient register looked like an old gas pump register with the dials spinning asthe quotient grew.

A friend of mine in high school rented a mechanical calculator to do lots of arithmetical homework.On the front in bold letters it said "DO NOT DIVIDE BY ZERO!" Of course, that was an invitation.So about half way through his assignment he starts a division by zero. The calculator was runon electricity, so it just kept spinning the digits in the quotient. Quite fun to watch, but tiresomeafter a while. So he punched "clear" and all the other buttons and levers he could find. Nothingstopped the process. Finally he got the idea to pull the plug, and presto! it stopped.

So he thought that he'd better get back to his homework. He plugged it back in and presto! itresumed the division calculation. He did the rest of his assignment by hand. It appears thatsomething had to be reset internally to get it to stop the division. I suspect when he returnedthe machine he just set it on the counter and quickly headed for the door!

The old electric mechanical calculators (which were made of metal and weighed about 60 lbs)demonstrated quite well the "repeated subtraction" algorithm for division. You could watch itas it proceeded through a calculation as the dials spun and the carriage shifted. Slow by today'sstandards, but it got the job done (if not dividing by zero!).

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

Re: Continuous Functions

Okay, I understand the fact that you can not divide x by 0, but i was strictly talking about 0 / 0, not 1, 2, or any other number. I don't understand why you can't divide by 0 in that case, because I do see 0 x 0 = 0 if u changed it to multiplication. Also, I didn't realize that rule applied to 0 as well, that was why I was asking. I already knew about stuff like 1/0 is undefined though and already understand why that is. Also I have read every post, and it still doesn't seem to prove to me that 0 / 0 is undefined, maybe because I'm looking at it the wrong way? Though I also worry about arguing this further, as this might only end up being something just like 0.999...=1. So, I'm just going to assume its a rule right now I don't... personally agree with. If there's better reasoning for it though, I would not mind hearing it.

There are always other variables. -[unknown]But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -AristotleEverything makes sense, one only needs to figure out how. -[unknown]

Re: Continuous Functions

Actually, I just realized I was looking at it differently. This in my opinion is another futile argument, I understand why it's a rule now. No need to further explain to me. I also understand the 2 = 1 proof now and see the error, I was having a...similar kind of issue with it. So with all my questions answered, I go back to with what I was originally going to say, but didn't say it quite clearly. I think the continuous functions page is very clear and well done. I didn't have an issue understanding it, only understanding indirectly relating things.

There are always other variables. -[unknown]But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -AristotleEverything makes sense, one only needs to figure out how. -[unknown]

Re: Continuous Functions

I'd agree, because I have high emphasis on myself for understanding things. It really bothers me if I walk away from something without understanding it well. It has also done nothing but help me with...well pretty much everything I know now (though there is still a lot I don't understand). So yeah, I will often question something I don't understand to better understand it.

There are always other variables. -[unknown]But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -AristotleEverything makes sense, one only needs to figure out how. -[unknown]

Re: Continuous Functions

Hi MIF

You can put on the same page the definition of a smooth function.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.